CHAP. Hi] SYMMETRIC FUNCTIONS MODULO p. 99
It is shown that (l(p))n equals the number of combinations of n+p — 1 things p — 1 at a time. Various algebraic relations between binomial coefficients are derived.
L. Gegenbauer285 considered the polynomial
p-2+k .
and proved that
—bp_2 (mod p),
p~ =—frp-2~~&2p~3 (mod p),
and deduced the theorem on the divisibility of sn by p.
E. Lucas286 proved the theorem on the divisibility of sn by p by use of the symbolic expression (s+l)n—sn for rcn —1.
N. Nielsen28601 proved that if p is an odd prune and if A; is odd and l<k<p — 1, the sum of the products of 1,..., p—1 taken k at a time is divisible by p2. For fc=p—2 this result is due to Wolstenholme.257
N. M. Ferrers287 proved that, if 2n+l is a prime, the sum of the products of 1, 2,..., 2n taken r at a time is divisible by 2n+l if r<2n [Lagrange18], while the sum of the products of the squares of 1,..., n taken r at a time is divisible by 2n+l if r<n. [Other proofs by Glaisher.294]
J. Perott288 gave a new proof that sn is divisible by p if n<p — 1.
R. Rawson289 proved the second theorem of Ferrers.
G. Osborn290 proved for r<p —1 that sr is divisible by p if r is even, by p2 if r is odd; while the sum of the products of 1,..., p —1 taken r at a time is divisible by p2 if r is odd and Kr<p.
J. W. L. Glaisher291 stated theorems on the sum Sr(ai,. .., a,-) of the products of ai,..., a* taken r at a time. If r is odd, Sr(l,..., n) is divisible by n+1 (special case n+l a prime proved by Lagrange and Ferrers). If r is odd and > 1, and if n+l is a prime> 3, jSr(l,. .., n) is divisible by (n+l)2 [Nielsen286"]. If r is odd and >1, and if n is a prime >2, Sr(l,.. ., n) is divisible by n2. If n+l is a prime, Sr(l2,. .., n2) is divisible by n+l for r = l,.,.,n — 1, except for r=n/2, when it is congruent to (— l)1+n/2 modulo n+l. If p is a prime ^n, and k is the quotient obtained on dividing n+l by p, then Sp-i(l,. .., n)=— k (mod p); the case n = p —1 is Wilson's theorem.
286Sitzungsber. Ak. Wiss. Wien (Math.), 95 II, 1887, 616-7.
18flTh6orie des nombrcs, 1891, 437.
28<*>Nyt Tidsskrift for Mat., 4, B, 1893, 1-10.
'"Messenger Math., 23, 1893-4, 56-58.
""Bull, des sc. math., 18, I, 1894, 64. Other proofs, Math. Quest. Educ. Times, 58, 1893, 109;
4, 1903, 42.
""Messenger Math., 24, 1894-5, 68-69. »°Jbid., 25, 1895-6, 68-69. ™Ibid., 28, 1898-9, 184-6. Proofs2".